Slow-roll, acceleration, the big rip and the Wentzel–Kramers–Brillouin approximation inthe non-linear Schrödinger-type formulation of scalar field cosmology

Abstract

Aspects of the non-linear Schrödinger (NLS) type of formulation of scalar (phantom) fieldcosmology for slow-roll, acceleration, the Wentzel–Kramers–Brillouin (WKB)approximation and the big rip singularity are presented. Slow-roll parameters for thecurvature and barotropic density terms are introduced. We re-express all slow-rollparameters, slow-roll conditions and the acceleration condition in NLS form. The WKBapproximation in the NLS formulation is also discussed while simplifying to the linear case.Most of the Schrödinger potentials in the NLS formulation are very slowly varying; hencethe WKB approximation is valid in some ranges. In the NLS form of the bigrip singularity, two quantities are infinite instead of three. We also found thatapproaching the big rip, , which is the same as the effective phantom equation of state in the flat case.